Determine whether the following
numbers are prime, composite, or neither. So just as a bit of
review, a prime number is a natural number-- so one of
the counting numbers, 1, 2, 3, 4, 5, 6, so on and so forth--
that has exactly two factors. So its factors are 1 and itself. So an example of a
prime factor is 3. There's only two
natural numbers that are divisible into 3-- 1 and 3. Or another way to think about
it is, the only way to get 3 as a product of other
natural numbers is 1 times 3. So it only has 1 and itself. A composite number
is a natural number that has more than just
1 and itself as factors. And we'll see examples of
that and neither-- we'll see an interesting case
of that in this problem. So first let's think about 24. So let's think
about all of the-- I guess you could think of
it as the natural numbers or the whole numbers,
although 0 is also included in whole numbers. Let's think of all of the
natural counting numbers that we can actually
divide into 24 without having any remainder. We'd consider those the factors. Well, clearly it is
divisible by 1 and 24. In fact, 1 times
24 is equal to 24. But it's also divisible by 2. 2 times 12 is 24. So it's also divisible by 12. And it is also divisible by 3. 3 times 8 is also equal to 24. And even at this point,
we don't actually have to find all of the factors
to realize that it's not prime. It clearly has more factors
than just 1 and itself. So then it is clearly
going to be composite. This is going to be composite. Now, let's just finish factoring
it just since we started it. It's also divisible by 4. And 4 times 6-- had just
enough space to do that. 4 times 6 is also 24. So these are all of the
factors of 24, clearly more than just one and 24. Now let's think about 2. Well, the non-zero whole numbers
that are divisible into 2, well, 1 times 2
definitely works, 1 and 2. But there really
aren't any others that are divisible into 2. And so it only has two
factors, 1 and itself, and that's the definition
of a prime number. So 2 is prime. And 2 is interesting because it
is the only even prime number. And that might be
common sense you. Because by definition, an
even number is divisible by 2. So 2 is clearly divisible by 2. That's what makes it even. But it's only
divisible by 2 and 1. So that's what makes it prime. But anything else
that's even is going to be divisible by
1, itself, and 2. Any other number
that is even is going to be divisible by
1, itself, and 2. So by definition,
it's going to have 1 and itself and something else. So it's going to be composite. So 2 is prime. Every other even number
other than 2 is composite. Now, here is an
interesting case. 1-- 1 is only divisible by 1. So it is not prime,
technically, because it only has 1 as a factor. It does not have two factors. 1 is itself. But in order to
be prime, you have to have exactly two factors. 1 has only one factor. In order to be
composite, you have to have more than two factors. You have to have 1, yourself,
and some other things. So it's not composite. So 1 is neither
prime nor composite. And then finally we get to 17. 17 Is divisible by 1 and 17. It's not divisible by 2,
not divisible by 3, 4, 5, 6. 7, 8, 9 10, 11, 12,
13, 14, 15, or 16. So it has exactly two
factors-- 1 and itself. So 17 is once
again-- 17 is prime.